Deformations of viscoelastic materials such as soft tissues, metals at high temperature, and polymers can be described as Volterra integral equations of the second kind. We consider the viscoelasticity model problem involving with \textit{Dirichlet Prony} series kernel, which resulting constitutive relation with exponentially decaying faded memory. We introduce \textit{internal variables} to replace Volterra integral to avoid the use of numerical integration for the convolution. We can deal with the fading memory by solving auxiliary ordinary differential equation systems govern by internal variables. We use a spatially discontinuous Galerkin finite element method and a finite difference method in time to formulate the fully discrete problem. We present \textit{a priori} analysis for long time viscoelastic response without Gr\"onwall's inequality. At the end, we carry out a number of numerical experiments based on using the FEniCS environment, \texttt{https://fenicsproject.org/}.
翻译：软组织、高温金属和聚合物等粘结性材料的变形可称为第二类的Volterra整体方程式。 我们考虑与\ textit{ Drichlet Prony} 系列内核有关的粘结性模型问题, 由此形成与指数衰减的记忆淡化的关系。 我们引入\ textit{ 内部变量} 取代Volterra 整体, 以避免使用数字集成, 我们可以通过解决由内部变量调节的辅助性普通差分方程式系统来处理衰落的记忆。 我们使用空间不连续的Galerkin定式元素法和固定差异法来及时制定完全离散的问题。 我们提出\ textit{ a priti} 分析长期的粘结性反应, 不使用 Gr\ “ onward” 的不平等。 最后, 我们根据FENCS环境,\ texttt{s://fencsproject.org/}
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